Boundary Conditions
Much of seismic prospecting is based on computer processing seismic data to migrate the data to form a true image of the subsurface or to infer a physical property model of the subsurface through data inversion. Migration and inversion cannot be performed analytically, and therefore must be performed using numerical methods on a computer. The most efficient migration method is reverse time migration (RTM). Both RTM and inversion require model simulation of predicted/measured seismic data, where the model is a model of subsurface velocity or other physical property affecting propagation of seismic waves. In numerical simulation of seismic data (sometimes referred to herein simply as data), large computational domains must be truncated to fit into the computer's memory. Artificial boundaries are introduced by this process. The reflections from an artificial, non-physical boundary may possibly bring artifacts into the image. Correct and suitable implementations of the boundaries are among the major problems of the numerical simulations. There are several different methodologies to deal with the problem, which include: 1) Non-reflecting boundary conditions (which will not be discussed further in this document); 2) Absorbing Boundary Conditions (“ABC”), see for example Kosloff et al. (1986); and 3) perfectly matched layers (“PML”) by Berenger (1994).
Absorbing Boundary Conditions and the Perfectly Matched Layers
Absorbing Boundary Conditions were introduced by Kosloff et al. (1986). The term “ABC” as used herein shall be understood to refer to the boundary conditions according to the Kosloff (1986) reference. This is an unconditionally stable method with relatively good absorption properties. The drawback of the method is coherent reflections from the fixed surfaces, like an air/water interface, which could build an artifact in the image. To improve absorption, one might need to increase the number of absorbing layers (referred to as padding), which in turn will negatively affect computational efficiency. Moreover, very low frequency reflections will still be a problem because absorption is a function of the number of wavelengths in the absorbing zone.
The perfectly matched layers (“PML”) absorbing boundary condition by Berenger (1994) is another commonly used way to approximate the radiation boundary condition for the sides and bottom of an earth model where the earth model is assumed to have infinite extent but the computational model has finite extent. Up to the discretization error, waves will not reflect from external boundaries of the computational model that are designated to have the radiation boundary condition. In media where the method is stable (see below), PML will give perfect results.
In the standard form of PML as described by Marcinkovich and Olsen (2003), every derivative normal to an exterior boundary has a wave field dissipation operator applied. Several issues arise with standard PML (sometimes referred to as 1D PML) operators. For general anisotropy, if the group velocity and the phase velocity have different signs for the direction normal to the boundary, PML becomes unstable and energy can be amplified rather than attenuated at the boundary (Bechache et al., 2001, and Loh et al., 2009, and Oskooi and Johnson, 2011). Stable and efficient PML implementation is still an active area of research. There are many methods suggested to address the stability issue—multiaxial-PML (M-PML) by Dmitriev et al., 2011, convolutional-PML (CPML) with complex shift by Zhang et al., 2010, and more recent developments—coordinates stretching in the PML region by Duru et. al., 2014. Note that all these methods have high cost and possibly degrade the effectiveness of the absorption. Moreover, the fundamental problem of the stability for these methods is not fully resolved.
In its general meaning, the term absorbing boundary conditions embraces both PML boundary conditions and ABC boundary conditions. To avoid terminology confusion herein, the term ABC will refer only to the Kosloff-type absorbing boundary conditions.
To summarize some drawbacks of existing methods for handling boundary conditions for computational domains used in model simulation of seismic data:                1. ABC—stable, but not effective in absorption.        2. Standard PML—almost perfect absorption, but stability is compromised for some anisotropic materials.        3. M-PML—high cost, and there is compromise between quality and stability; difficult to find optimal damping. Absorption is not very effective.        4. Coordinate stretching in the PML region—high cost, with a compromise between quality and stability; difficult to find optimal damping.        5. Complex shift—see item 2 above; the fundamental problem of stability is not fully resolved.What is needed is a method that provides a better compromise between absorption, stability, and cost. The present invention satisfies this need.        